The sets $V_\lambda$ for limit ordinals $\lambda>\omega$ are precisely the models of second-order Zermelo set theory (including foundation scheme, infinity, and choice) plus the cumulative hierarchy axiom CHA, which asserts that every set $x$ is in some $V_\beta$, where this is a set for which there is well-ordered sequence $\langle V_\alpha\mid\alpha\leq\beta\rangle$ obeying the recursive definition of the cumulative hierarchy. 

It is easy to see that every $V_\lambda$ satisfies that theory, and conversely, if a model satisfies this theory, then because we have the second-order separation axiom in second-order Zermelo, it will be correct about power sets, and so the cumulative hierarchy that it builds will be correct. So the model will be (isomorphic to) $V_\lambda$ for some limit ordinal $\lambda>\omega$.

This theorem, which was observed by an undergraduate student of mine Donghui Jia in Oxford last term, who wrote on it as a version of Zermelo's quasi-categoricity theorem, in which Zermelo proved that the models of second-order ZFC are precisely $V_\kappa$ for inaccessible cardinals $\kappa$.  

Meanwhile, you had asked for consequences of replacement that hold in every $V_\lambda$, and the cumulative hierarchy axiom is one. Mathias constructed supertransitive models of Zermelo set theory in which $V_\omega$ does not exist. Modifications of the Mathias slim-model technique allow one to construct models of Zermelo set theory where the first omitted $V_\lambda$ is for any desired limit ordinal $\lambda$, even though the model has order-types much exceeding $\lambda$. I believe also that there are models of Zermelo set theory that do not satisfy transitive containment---that is, not every set has a transitive closure; this holds in every $V_\lambda$, but one uses replacement to prove it.

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**Warning.** To get the CHA, it isn't quite enough to say that for every ordinal $\beta$ there is a set $V_\beta$ arising from a sequence $\langle V_\alpha\mid\alpha\leq\beta\rangle$ obeying the recursive definition of the cumulative hierarchy. The reason is that the ordinals might run out before you expect, and perhaps you have $V_\alpha$ for all the ordinals $\alpha$ that exist, but there are still more sets of higher rank, whose ordinals do not exist.